How dense is $X=\{2^m3^n:m,n\in\Bbb Z\}$ in $\Bbb R^+$?
I could write the question more precisely but this would probably be counterproductive as someone answering this question would probably know better than I how to do so.
I guess I might ask if $x$ is some arbitrary real number, does every arbitrarily small ball around it contain some value in $X$?
I have a hunch based upon some prior reading that this is related to Baker's theorem, but this may be misplaced.
I didn't know what to tag - so please do edit.
Details of Daniel Fisher comment.
$Y=\{m \log 2 +n \log 3 \mid (m,n) \in \mathbb Z^2\}$ is an additive subgroup of $\mathbb R$. Hence it is either dense or discrete (as all additive subgroup of the reals is).
If $Y$ was discrete, it would exists $a > 0$ such that for all $y \in Y$, it exists $p \in \mathbb Z$ such that $y = pa$. And $\log 2, \log 3 \in Y$ implies that $\frac{\log 2}{\log 3}$ is rational and therefore that it exists $(r,s) \in \mathbb Z^2 \setminus \{(0,0)\}$ such that $2^r3^s =1$. That can't be as $2,3$ are coprime numbers.
Finally $Y$ is dense in $\mathbb R$ and $X$ dense in $\mathbb R_+$ as $x \mapsto e^x$ is continuous.